Question

\[\Psi(x,t) = Ae^{ -a\left[\frac{mx^{2}}{\hbar}+it\right]}\]\[\langle p^2 \rangle =?\]\[A=\sqrt[4]{\frac{2am}{\pi\hbar}}\]\[\frac{\partial^2\Psi}{\partial x^2} =\left(\frac{-2ma}{\hbar}+ \frac{4m^2a^2x^2}{\hbar^2}\right)\Psi\]

Answer 1

\[\langle p^2 \rangle = \int\limits_{-\infty}^{\infty}\Psi^*\left(\frac{\hbar}{i}\frac{\partial}{\partial x}\right)^2 \Psi~\text d x\]
\[= \left(\frac{\hbar}{i}\right)^2\int\limits_{-\infty}^{\infty}\Psi^*\frac{\partial^2\Psi}{\partial x^2}~\text d x\]
\[= -\hbar^2\int\limits_{-\infty}^{\infty}\Psi^*\left(\frac{-2ma}{\hbar}+ \frac{4m^2a^2x^2}{\hbar^2}\right)\Psi ~\text d x\]

Answer 2

i believe this is the font from old windows called "wingdings"

Answer 3

\[= -\hbar^2\left[\frac{-2ma}{\hbar}\int\limits_{-\infty}^{\infty}|\Psi|^2\text dx+ \frac{4m^2a^2}{\hbar^2}\int x^2|\Psi|^2 ~\text d x\right]\]

Answer 4

\[= {2ma}{\hbar}\int\limits_{-\infty}^{\infty}|\Psi|^2\text dx- {4m^2a^2}\int x^2|\Psi|^2 ~\text d x\]\[={2ma}{\hbar}(1)- {4m^2a^2}\frac{\hbar}{4am} \]\[={2ma}{\hbar}- ma\hbar\]\[=ma\hbar\]

Answer 5

\[\color{purple}\checkmark\]

Answer 6

These all seem Greek to me.

Answer 7

only the Psi \(\Psi\) is a greek letter, oh and pi \(\pi\)

Answer 8

\[\hbar=\frac h{2\pi}\]